For most of my life, one single (family of) Diophantine equation(s) dominated the list of the world's most celebrated unsolved mathematical problems. Perhaps the world we live in now has grown too sophisticated for such heavy focus on such a narrow question (visions of Fermat's MO postings getting closed as "Too localized"), but if not:
What specific unsolved Diophantine equations would today's number theorists most like to crack? (And why - historical provenance, application to another part of mathematics, "test" question for a major arithmetical theory, etc.)
"Specific" means, for example, "solve BSD, etc. to find an algorithm to decide the solvability of all elliptic curves" doesn't count, nor is this is the place to talk about anything like the $abc$ conjecture.
Answers should not depend on any sort of coding, Matiyasevich-style.